Method of making a motor with reduced cogging torque

ABSTRACT

To reduce the cogging torque of servomotors, electric power steering motors, and others, there is provided a permanent magnet motor comprising: a rotor  10  comprising a rotor yoke  11  and a plurality of permanent magnets (M 1 -M 10 ); and a stator  20  comprising a stator yoke  22 , salient magnetic poles  21 , and armature windings  23 , wherein at least one of the permanent magnets is disposed in an adjustment position that is displaced from a corresponding reference position in at least one of the circumferential, radial, and axial directions of the rotor yoke, and the plurality of permanent magnets excluding the permanent magnet disposed in the adjustment position is disposed in the reference positions, and wherein the adjustment position is set so that the permanent magnet motor in which at least one of the plurality of permanent magnets is disposed in the adjustment position has a smaller cogging torque than a permanent magnet motor in which all of the plurality of permanent magnets are disposed in the reference positions; and a method for adjusting a cogging torque of a permanent magnet motor.

CROSS-RELATED APPLICATIONS

This application is a divisional application of U.S. patent applicationSer. No. 11/205,523 filed Aug. 17, 2005, now allowed, which claimspriority from Japanese Patent Application No. 2004-240222; filed Aug.20, 2004, the disclosures of which are incorporated herein by referencein their entireties.

BACKGROUND OF THE INVENTION

The present invention relates to permanent magnet motors, andspecifically to permanent magnet motors for servomotors and electricpower steering motors etc., with reduced cogging torque.

There has been a demand for miniaturized and low-loss servomotors andelectric power steering motors, and by employing a motor in whicharmature windings are concentratedly wound around respective salientmagnetic poles of a stator, the amount of the windings that extendbeyond an end portion of the stator is reduced, and thus the length ofthe motor and copper loss in the windings are reduced. Also, there hasbeen a strong demand for reducing the cogging torque of motors in orderto improve positioning accuracy and to reduce noise and vibrations.

As a method for reducing the cogging torque of a concentrated windingpermanent magnet motor, Japanese Patent No. 2135902, which is hereinincorporated by reference, describes a method in which the relationshipbetween the number P of poles of a rotor and the number M of salientstator poles is made to satisfy P=6n±2 and M=6n, wherein n is an integerof 2 or more, or P=3m±1 and M=3m, wherein m is an odd number of 3 ormore. The permanent magnet motor has a cogging torque that pulsates, thenumber of which is the least common multiple of the number P of polesand the number M of salient poles per revolution of the rotor, and themagnitude of the cogging torque decreases as the number of pulsationsincreases. Based on this principle, the permanent magnet motor has thecombination of the number P of poles and the number M of salient polessuch that the least common multiple of the number P of poles of therotor and the number M of salient poles of the stator increases underthe condition that a three-phase winding connection is possible. Table 1shows a summary of specific combinations of the number P of poles andthe number M of salient poles.

TABLE 1 Number of Number of Number of cogging poles P salient poles Mtorque pulsations n 2 10, 14 12 60, 84 3 16, 20 18 144, 180 m 3  8, 10 972, 90 5 14, 16 15 210, 240

A permanent magnet motor having the combination of P=10 and M=12 shownin Table 1 was designed. FIG. 11 shows a schematic cross-sectional viewof the permanent magnet motor having 10 poles and 12 slots, in a planeperpendicular to the axis of the permanent magnet motor. In FIG. 11, themotor includes: a rotor 10 in which ten neodymium magnets (M1 to M10)having a residual magnetic flux density of 1.26 Tesla are disposed on arotor yoke 11 made of low carbon steel S45C at even intervals such thatthe polarities of the magnets alternate in the circumferentialdirection; and a stator 20 having a stator yoke 22 that is opposed tothe permanent magnets and that is formed from an isotropic silicon steelsheet 35A300 on which twelve salient magnetic poles 21 are disposed ateven intervals in the circumferential direction, ten-turn armaturewindings 23, which are wound around the magnetic poles and which areserially connected in each of the U, V and W phases that are three-phaseconnected. In FIG. 11, the direction of magnetization of the permanentmagnets is indicated by an arrow in each permanent magnet. Moreover, atthe center of the rotor, the direction of revolution of the rotor isindicated by an arrow. The permanent magnets have a shape in which thethickness of the permanent magnets is reduced at both end portions. FIG.12 shows a specific shape of the permanent magnets. In FIG. 12, Ri=23mm, Ro=10 mm, D=15 mm, and W=12 mm. By reducing the thickness of apermanent magnet at both end portions, the distribution of the magneticflux density at an air gap is smoothed, and thus an effect of reducingthe cogging torque is achieved. The rotor and the stator have a lengthof 40 mm in the axial direction, and the air gap between the rotormagnets and the stator magnetic poles is 1 mm. In FIG. 11, symbols shownin the windings indicate the directions of the windings: a solid circle() indicates the direction emerging from the paper surface and a cross(×) indicates the direction entering the paper surface. The rated torquewas set to be 2 Nm (newton meter) when the motor is driven with asinusoidal current having an effective value of 20 A (ampere).

The cogging torque of the above-described permanent magnet motor wascalculated assuming that there was no variation in the characteristicsand the dimensions of the magnets and that all factors were in an idealstate. It was found that the cogging torque had a waveform having a peakvalue of 0.0003 Nm and 60 pulsations per revolution of the rotor. Sincethe cogging torque is expressed as the difference between the maximumand the minimum points of the waves, in this case, the cogging torquewas 0.0006 Nm, which was 0.03% of the rated value, that is, a very smallvalue. In the case of permanent magnet motors such as for electric powersteering, a weak cogging torque affects the steering feel, so it isdesirable that the cogging torque is no greater than 0.5% of the ratedtorque (in the present case, no greater than 0.01 Nm).

Next, as Comparative Example 1, the designed motor was actuallyfabricated, and the cogging torque of that motor was measured. Neodymiummagnets were used as the permanent magnets, which were made by filling adie having a Japanese roof tile-like shape with magnet powder, pressingthe die in a transverse magnetic field, sintering the pressed magneticpowder, and subjecting the sintered product to heating and which werethen ground to precision of 0.05 mm or less using a whetstone. Moreover,a dedicated jig was prepared to position the permanent magnets on therotor yoke, and the positioning was performed with a precision of 0.05mm or less. The stator yoke was made by laser cutting out pieces of 0.35mm silicon steel sheet to a predetermined shape, and laminating thepieces with a laminating method, referred to as “parallel laminatingmethod”, in which the pieces are laminated with their rolling directionsin a uniform direction. After the lamination, the pieces were fastenedat eight points on a peripheral portion of the stator yoke by laserwelding, and the inner surface of the stator yoke opposed to thepermanent magnets was ground to increase dimensional accuracy.

FIG. 13 shows an actually-measured waveform of the cogging torque of thepermanent magnet motor according to Comparative Example 1(parallel-laminated stator). The measured cogging torque waveformaccording to Comparative Example 1 was a waveform having 10 pulsationsper revolution of the rotor. The entire waveform is shifted to thenegative side due to a force working against revolution, which is called“loss torque”. This is caused by the hysteresis loss of the stator yoke.Since the cogging torque is expressed as the difference between themaximum and the minimum points of the waves, the cogging torque ofComparative Example 1 shown in FIG. 13 was 0.0274 Nm. FIG. 14 shows theresults of a Fourier analysis in which the cogging torque waveformaccording to Comparative Example 1 was divided into components ofrespective orders of the waveform. Here, “order” is the number ofpulsations that appear during one revolution of the rotor. For example,“duodenary component” is a component having twelve pulsations during onerevolution of the rotor. Regarding the components of the cogging torqueof Comparative Example 1 shown in FIG. 13, the denary component is0.0061 Nm, the duodenary component is 0.0077 Nm, the vigenary componentis 0.0016 Nm, and the twenty-fourth order component is 0.0007 Nm. Itshould be noted that the order components shown in FIG. 14 represent thepeak values of the respective order components, which are values halfthe cogging torque. Checking the order components of the cogging torquewaveform makes it possible to know what the cogging torque is attributedto. In the case of the permanent magnet motor according to ComparativeExample 1, it is believed that the denary, vigenary, and thirtieth ordercomponents, whose orders correspond to multiples of the number of poles,are caused by variations in the stator yoke, and the duodenary,twenty-fourth order, and thirty-sixth order components, whose orderscorrespond to multiples of the number of salient poles, are caused byvariations in the permanent magnets. If there is no variation in thepermanent magnets, the positional relationship between the stator yokeand the permanent magnets has rotational symmetry with respect to themagnetic pole pitch angle, so that the cogging torque caused byvariations in the stator yoke has a waveform in which the number ofcycles of the fundamental wave corresponds to the number of poles perrevolution. Similarly, if there is no variation in the stator, thepositional relationship between the stator and the magnets hasrotational symmetry with respect to the salient pole pitch angle, sothat the cogging torque caused by variations in the magnets has awaveform in which the number of cycles of the fundamental wavecorresponds to the number of salient poles per revolution. The sixtiethorder, which corresponds to the least common multiple of the number ofpoles and the number of salient poles, is the cogging torque when thepermanent magnet motor was an ideal permanent magnet motor. The coggingtorque of Comparative Example 1 shown in FIG. 13 has large components oforders of multiples of 10. Since even an isotropic steel sheet has somemagnetic anisotropy in the rolling direction, when the stator yoke ismade by the parallel laminating method in which rolling directions ofthe pieces are made uniform, the magnetic anisotropy remains. It appearsthat the influence of this residual magnetic anisotropy causes thecogging torque.

In order to eliminate the magnetic anisotropy of the stator yoke, in“Measurement of Cogging Torque of Permanent Magnet Motor Due to MagneticAnisotropy of Non-oriented Electrical Steel Sheet”, Proceedings of theNational Convention of the Institute of Electrical Engineers of Japan,5-016., which is herein incorporated by reference, a method in whichlamination is performed such that the rolling direction successivelyrotates, which is called “rotational laminating method”, is used. AsComparative Example 2, a permanent magnet motor having arotational-laminated stator was fabricated. The rotor and the method formaking the stator yoke except for the laminating method were the same asin Comparative Example 1. FIG. 15 shows an actually-measured waveform ofthe cogging torque of the permanent magnet motor according toComparative Example 2 (rotational-laminated stator). Moreover, FIG. 14also shows the results of a Fourier analysis in which the cogging torquewaveform according to Comparative Example 2 was divided into componentsof respective orders of the waveform. In Comparative Example 2, ascompared to Comparative Example 1, the denary, vigenary, and thirtiethorder components were significantly reduced to less than 0.0002 Nm, andthe cogging torque was 0.0122 Nm. In this way, the cogging torqueattributed to the stator yoke can be reduced.

However, even with such a method, the duodenary, twenty-fourth order,and thirty-sixth order components, which are attributed to the permanentmagnets, cannot be eliminated. In particular, when the permanent magnetmotor is used for electric power steering motors, lower order componentsgreatly affect the steering feel, so that it is desirable to reduce theduodenary component.

In order to eliminate an order component (duodenary component, in thecomparative example) of the cogging torque caused by the permanentmagnets, a method has been used in which each of the permanent magnetsof the rotor are divided into a plurality of magnets in the axialdirection and are skewed. In this case, the skew angle is 360°/thenumber of salient poles (30°, in the comparative example). However, inthe case of the permanent magnet motor in which a reduced cogging torqueis achieved by using any of the combinations of the number of poles andthe number of salient poles shown in Table 1, the difference between thenumber of poles and the number of salient poles is small, so that whenskewing is performed, different magnetic poles come to be opposed to onesalient pole at the same time, and thus driving torque is not generated.Accordingly, there has been no effective measure for reducing suchcogging torque caused by permanent magnets, and there has been a problemin that there are practical limitations to industrial processes forreducing the cogging torque by suppressing variations in the permanentmagnets.

SUMMARY OF THE INVENTION

It is an object of the present invention to reduce the cogging torque ofservomotors, electric power steering motors, and others, and to providea permanent magnet motor with which it is possible to realize a robot inwhich a servomotor that is capable of positioning with high accuracy isused and an electric power steering that generates low vibrations andprovides good steering feel by eliminating a component of the toggingtorque caused by permanent magnets.

According to one aspect of the present invention, there is provided apermanent magnet motor comprising:

a rotor comprising a rotor yoke and a plurality of permanent magnetsthat are disposed on a side face of the rotor yoke at predeterminedintervals such that polarities of the permanent magnets alternate in acircumferential direction of the rotor yoke; and

a stator comprising a stator yoke that is disposed at a distance fromthe rotor, salient magnetic poles that are disposed on the stator yokeat even intervals with respect to a circumferential direction and thatare opposed to the permanent magnets, and three-phase connected armaturewindings that are concentratedly wound around the respective salientmagnetic poles,

wherein at least one of the permanent magnets is disposed in anadjustment position that is displaced from a corresponding referenceposition in at least one of the circumferential, radial, and axialdirections of the rotor yoke, wherein the reference positions arelocated at even intervals with respect to the circumferential directionof the rotor yoke, are equidistant from the central axis with respect tothe radial direction, and are equidistant from axial direction ends ofthe rotor yoke with respect to the axial direction, and the plurality ofpermanent magnets excluding the permanent magnet disposed in theadjustment position is disposed in the reference positions, and

wherein the adjustment position is set so that the permanent magnetmotor in which at least one of the plurality of permanent magnets isdisposed in the adjustment position has a smaller cogging torque than apermanent magnet motor in which all of the plurality of permanentmagnets are disposed in the reference positions.

According to another aspect of the present invention, there is provideda method for adjusting a cogging torque of a permanent magnet motor, thepermanent magnet motor comprising:

a rotor comprising a rotor yoke and a plurality of permanent magnet thatare disposed on a side face of the rotor yoke at predetermined intervalssuch that polarities of the permanent magnets alternate in acircumferential direction of the rotor yoke; and

a stator comprising a stator yoke that is disposed at a distance fromthe rotor, salient magnetic poles that are disposed on the stator yokeat even intervals with respect to the circumferential direction and thatare opposed to the permanent magnets, and three-phase connected armaturewindings that are concentratedly wound around the respective salientmagnetic poles,

wherein the method comprises steps of:

disposing the plurality of permanent magnets in reference positions thatare located at even intervals with respect to the circumferentialdirection of the rotor yoke, are equidistant from the central axis withrespect to a radial direction, and are equidistant from axial directionends of the rotor yoke with respect to an axial direction, such that thepolarities of the permanent magnets alternate in the circumferentialdirection; and

moving at least one of the permanent magnets in at least one of thecircumferential, radial, and axial directions of the rotor yoke toadjust the cogging torque.

As will be described in detail below, according to the presentinvention, it is possible to reduce the cogging torque caused byvariations in the permanent magnets, and to realize a robot in which aservomotor that is capable of positioning with high accuracy is used andan electric power steering motor that generates low vibrations andprovides good steering feel.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a schematic front view of a rotor when viewed from adirection perpendicular to the axis of the rotor, according to the firstembodiment.

FIG. 2 shows a schematic cross-sectional view of a permanent magnetmotor in a plane passing through the axis of the rotor, according to thesecond embodiment.

FIG. 3 shows a schematic cross-sectional view of a permanent magnetmotor in a plane passing through the axis of the rotor, according to thethird embodiment.

FIG. 4 shows a schematic front view of a rotor when viewed from adirection perpendicular to the axis of the rotor, according to a fourthembodiment.

FIG. 5 shows the results of the Fourier analysis in which the waveformof the amount of the change in the cogging torque when the permanentmagnet was moved in the circumferential direction was divided intocomponents of respective orders of the waveform.

FIG. 6 shows the actually-measured waveforms of the amount of change inthe cogging torque when the permanent magnets in different magnetpositions were moved in the radial direction.

FIG. 7 shows the actually-measured waveform of the cogging torque of thepermanent magnet motor according to Working Example 1.

FIG. 8 shows the results of the Fourier analysis in which the waveformof the cogging torque according to Comparative Example 2 and WorkingExample 1 was divided into components of respective orders of thewaveform.

FIG. 9 shows the results of the Fourier analysis in which the waveformof the amount of change in the cogging torque when the permanent magnetwas moved in the radial direction was divided into components ofrespective orders of the waveform.

FIG. 10 shows the results of the Fourier analysis in which the waveformof the amount of change in the cogging torque when the permanent magnetwas moved in the axial direction was divided into components ofrespective orders of the waveform.

FIG. 11 shows a schematic cross-sectional view of the permanent magnetmotor having 10 poles and 12 slots, in a plane perpendicular to the axisof the permanent magnet motor, according to the working and comparativeexamples.

FIG. 12 shows a specific shape of the permanent magnets according to theworking and comparative examples.

FIG. 13 shows an actually-measured waveform of the cogging torque of thepermanent magnet motor according to Comparative Example 1(parallel-laminated stator).

FIG. 14 shows the results of a Fourier analysis in which the coggingtorque waveform according to Comparative Example 1 was divided intocomponents of respective orders of the waveform.

FIG. 15 shows an actually-measured waveform of the cogging torque of thepermanent magnet motor according to Comparative Example 2(rotational-laminated stator).

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

The present invention now will be described more fully hereinafter inwhich embodiments of the invention are provided with reference to theaccompanying drawings. This invention may, however, be embodied in manydifferent forms and should not be construed as limited to theembodiments set forth herein; rather, these embodiments are provided sothat this disclosure will be thorough and complete, and will fullyconvey the scope of the invention to those skilled in the art.

The terminology used in the description of the invention herein is forthe purpose of describing particular embodiments only and is notintended to be limiting of the invention. As used in the description ofthe invention and the appended claims, the singular forms “a”, “an” and“the” are intended to include the plural forms as well, unless thecontext clearly indicates otherwise.

Unless otherwise defined, all technical and scientific terms used hereinhave the same meaning as commonly understood by one of ordinary skill inthe art to which this invention belongs.

As in the foregoing description, according to the present invention,there is provided a permanent magnet motor comprising:

a rotor comprising a rotor yoke and a plurality of permanent magnetsthat are disposed on a side face of the rotor yoke at predeterminedintervals such that polarities of the permanent magnets alternate in acircumferential direction of the rotor yoke; and

a stator comprising a stator yoke that is disposed at a distance fromthe rotor, salient magnetic poles that are disposed on the stator yokeat even intervals with respect to a circumferential direction and thatare opposed to the permanent magnets, and three-phase connected armaturewindings that are concentratedly wound around the respective salientmagnetic poles,

wherein at least one of the permanent magnets is disposed in anadjustment position that is displaced from a corresponding referenceposition in at least one of the circumferential, radial, and axialdirections of the rotor yoke, wherein the reference positions arelocated at even intervals with respect to the circumferential directionof the rotor yoke, are equidistant from the central axis with respect tothe radial direction, and are equidistant from axial direction ends ofthe rotor yoke with respect to the axial direction, and the plurality ofpermanent magnets excluding the permanent magnet disposed in theadjustment position is disposed in the reference positions, and

wherein the adjustment position is set so that the permanent magnetmotor in which at least one of the plurality of permanent magnets isdisposed in the adjustment position has a smaller cogging torque than apermanent magnet motor in which all of the plurality of permanentmagnets are disposed in the reference positions.

In the present invention, the permanent magnet motor can have the sameconfiguration as conventional permanent magnet motors, except for itemsparticularly described below, such as the position of the permanentmagnets, magnet holding member, and others, so that a detaileddescription thereof is omitted. For the permanent magnets, in additionto a neodymium magnet, a samarium cobalt magnet, a ferrite magnet, and abonded magnet of these magnets may be used, and there is no particularlimitation regarding the material of the permanent magnets. Moreover, alow carbon steel or a silicon steel can be used for the rotor yoke. Therotor yoke can be a cylinder having a circular aperture in its center. Ashaft having the same diameter as the aperture is made to penetrate theaperture of the rotor yoke. A low carbon steel can be used for theshaft.

First, a plurality of permanent magnets is disposed in the referencepositions on the rotor yoke so that the polarities of the permanentmagnets alternate in the circumferential direction. Here, “referencepositions” are located at even intervals with respect to thecircumferential direction of the rotor yoke, are equidistant from thecentral axis with respect to the radial direction, and are equidistantfrom the axial direction ends of the rotor yoke with respect to theaxial direction. In other words, “reference positions” are the positionsin which the permanent magnets have been disposed in conventionalpermanent magnet motors. If there is no variation in the characteristicsand the dimensions of the magnets and all factors are in an ideal state,then when the permanent magnets are disposed in the reference positions,cogging torque can be minimized. The permanent magnets can be attachedto the rotor yoke by the attraction of the magnets.

Here, preferably, the plurality of permanent magnets is held in thereference positions by magnet holding member. The magnet holding memberscan hold the permanent magnets, preferably all of the permanent magnets,in the reference positions and in adjustment positions, as will bedescribed below, and the magnet holding member can include magnetholding yokes and magnet holders. The magnet holding yoke can be a ringhaving an aperture whose diameter is equal to that of the shaft.Furthermore, the magnet holding yoke has the same number of tap holes asthe permanent magnets, and the tap holes are disposed at even intervalsin the circumferential direction within a concentric circular shapewhose radius is equal to the distance of the axis of the rotor and thepermanent magnets. The periphery of the magnet holding yoke also canhave a polygonal shape. As the magnet holder, for example, a hexagonsocket set screw can be used. It is preferable to use a non-magneticmaterial that does not affect the magnetic field, such as aluminum,stainless steel, and resin materials such as MC nylon, for the magnetholding yokes and the magnet holders. First, the magnet holding yokesare fixed to the shaft respectively at both sides of the rotor yoke bybolting or adhesion. Then, both ends of each of the permanent magnetsthat are disposed in the reference positions on the surface of the rotoryoke are pressed down by the magnet holders provided for the magneticholding yokes, and thus the permanent magnets can be mechanically held.

Next, at least one of the permanent magnets is moved in at least onedirection of the circumferential direction, the radial direction, andthe axial direction of the rotor yoke, so as to adjust the coggingtorque. More specifically, the cogging torque is measured beforeadjustment of the cogging torque, that is to say, when the permanentmagnets are disposed in the reference positions, and then, the coggingtorque is measured after one of the permanent magnets is moved in atleast one direction of the circumferential direction, the radialdirection, and the axial direction of the rotor yoke, and furthermore,the cogging torque can be adjusted by determining based on the obtainedmeasurement values, which permanent magnet is to be moved and thedistance that the permanent magnet is to be moved using a linearprogramming method such that the cogging torque is more reduced. Inother words, according to another aspect of the present invention, thereis provided a permanent magnet motor in which at least one of permanentmagnets is disposed in an adjustment position that is displaced from acorresponding reference position in at least one of circumferential,radial, and axial directions of a rotor yoke, and the plurality ofpermanent magnets excluding the permanent magnet that is disposed in theadjustment position is disposed in the reference positions. Here, theadjustment position is set so that the permanent magnet motor in whichat least one of the plurality of permanent magnets is disposed in theadjustment position has a smaller cogging torque than a permanent magnetmotor in which all of the plurality of permanent magnets are disposed inthe reference positions.

That is to say, as will be described in detail below, according to thepresent invention, by moving a permanent magnet on the rotor yoke, anorder (duodenary, in the comparative example described above) componentof the cogging torque caused by the permanent magnets can besignificantly reduced.

When a permanent magnet on the rotor yoke is moved, the cogging torqueis changed. Here, as shown below, the components of the cogging torquethat changed are of the same orders (orders of multiples of 12, in thecomparative example) as the orders of the cogging torque caused by thepermanent magnets, and most of these components are of the smallestorder (duodenary, in the comparative example) of the relevant orders.Furthermore, the amount of change in the cogging torque is proportionalto the amount of movement. Moreover, the phase of the waveform of theamount of change in the cogging torque is changed by changing theposition (θ) of the magnet to be moved. By moving the permanent magnet,the component (duodenary component, in the comparative example) of thecogging torque caused by the permanent magnets is changed, and the phasethereof changes depending of the position of the magnet moved. Thisimplies that the cogging torque can be reduced by adjusting the positionof the magnet to be moved and the amount of movement of the permanentmagnet such that the waveform of the cogging torque caused by thepermanent magnets is canceled.

Accordingly, an analysis was carried out using a mathematicalprogramming method to determine which magnet should be moved and how farthat magnet should be moved in order to reduce the cogging torque. Here,since the amount of change in the cogging torque is proportional to theamount of movement, as described above, it is preferable to use a linearprogramming method as the mathematical programming method. Whenadjustment of the cogging torque is expressed as a linear programmingproblem, it can be expressed as “minimizing (1) under a condition (2)”,and by solving this problem, the position of the magnet to be moved andthe amount of movement of the permanent magnet can be determined. Itshould be noted that, the position of the magnet is used to specify eachof the plurality of permanent magnets disposed on the rotor yoke, andfor example, in the comparative example described above, there are tenpositions of the magnets, each of which represents the correspondingpermanent magnet.

$\begin{matrix}{Z = {{C_{0}{\sum\limits_{j = 1}^{B}\; X_{j}}} + {C_{1}W}}} & (1) \\\left. \begin{matrix}{{{\sum\limits_{j = 1}^{B}\; {a_{ij}X_{j}}} + \frac{W}{2}} \geq {{- T_{i}} - {\frac{1}{2}{tole}} + T_{l}}} \\{{{- {\sum\limits_{j = 1}^{B}\; {a_{ij}X_{j}}}} + \frac{W}{2}} \geq {T_{i} - {\frac{1}{2}{tole}} - T_{l}}} \\{X_{j},{W \geq 0}}\end{matrix} \right\} & (2)\end{matrix}$

In the formulae, the letters represent the following:

B represents the number of magnets.

represents the point at which the cogging torque is measured. That is tosay, each value of i represents an angle (0° to 360°) of revolution ofthe rotor, which is a point at which the cogging torque is measured.

j represents the position of the magnet. That is to say, j takes anyinteger between 1 and the number B of permanent magnets, and each valueof j represents a permanent magnet.

X_(j) is the amount of movement of the j-th magnet.

T_(i) is the cogging torque at the i-th point before adjustment of thecogging torque. Measurement data for the cogging torque beforeadjustment is input as T_(i).

a_(ij) is the amount of change in the cogging torque at the i-th pointper amount of movement of the j-th magnet, when the j-th magnet ismoved. More specifically, with respect to each i-th point, a_(ij) can beobtained by dividing the measurement value for the amount of change inthe cogging torque that is measured when the j-th magnet is moved by theamount of movement of the magnet.

T₁ is the loss torque before adjustment of the cogging torque. T₁ can bean average value of the measurement values of the cogging torque beforeadjustment.

tole is a target cogging torque. tole can be set as desired according tothe application of the permanent magnet motor. In particular, when thepermanent magnet motor is applied to servomotors, electric powersteering motors, and others, it is preferable that tole is set to nogreater than 0.5% of the rated torque.

W is the difference between the target cogging torque (tole) and acalculated value of the cogging torque after the permanent magnet hasbeen moved. More specifically, W is expressed by Formula (3) below:

$\begin{matrix}{W = {{\max\left( {T_{i} + {\sum\limits_{j = 1}^{B}\; {a_{ij}X_{j}}}} \right)} - {\min\left( {T_{i} + {\sum\limits_{j = 1}^{B}\; {a_{ij}X_{j}}}} \right)} - {tole}}} & (3)\end{matrix}$

C₀ and C₁ are coefficients, which are set according to the balancebetween the amount of movement of the magnet and the cogging torque. C₀is a number of 0 or more, and C₁ is a number more than 0. That is tosay, C₀ and C₁ are determined according to the balance of B, X_(j),tole, and W. If C₀ and C₁ are changed, then the solution of the linearprogram is changed, and thus calculations are performed while adjustingthese coefficients. An operator can obtain an optimum amount of movementX_(j) by performing a plurality of calculations while adjusting thevalues of C₀ and C₁ such that the target cogging torque can be achievedwith as small a total amount (ΣX_(j)) of movement of the magnets aspossible. More specifically, although it is not a particular limitation,a calculation is initially performed using C₀/C₁=0 to obtain the amountX_(j) of movement of each magnet. Here, when it is desired to reduce thetotal amount (ΣX_(j)) of movement of the magnets more, a furthercalculation is performed by increasing C₀/C₁, for example, usingC₀/C₁=0.001, to obtain the amount (X_(j)) of movement of each magnet. Byrepeating this process, it is possible to achieve the target coggingtorque with as small an amount of movement of the magnets as possible.

With the above-described linear programming problem, by inputting intothe linear program the data of the cogging torque before adjustment andthe data for the amount of change in the cogging torque when eachpermanent magnet is moved in, for example, the circumferentialdirection, the amount of movement of each permanent magnet can becalculated such that the cogging torque is reduced.

When the position of the magnet to be moved and the amount of movementof the permanent magnet are determined by the mathematical programmingmethod, the permanent magnet is moved in accordance with that result. Atthis time, the permanent magnet can be moved using a dedicated jig, aswhen the permanent magnet was disposed in the reference position.Moreover, it is preferable that the permanent magnet is moved with thesame precision as when the permanent magnet was disposed in thereference position. Moreover, after the permanent magnet is moved, it ispreferable that the plurality of permanent magnets is held in theadjustment positions by the magnet holding member.

Referring to FIG. 1, a first embodiment according to the presentinvention will be described. FIG. 1 shows a schematic front view of arotor when viewed from a direction perpendicular to the axis of therotor, according to the first embodiment. As described above, the rotor10 according to the first embodiment has a configuration in which at thefirst time a permanent magnet motor has a plurality of permanent magnets(M1 to M10) disposed in reference positions on a rotor yoke 11 throughwhich a shaft 12 passes, wherein the plurality of permanent magnets areheld in the reference positions from both sides of the rotor yoke 11 bymagnet holding member 13, including magnet holding yokes 14 and magnetholders 15, subsequently, one (M1) of the permanent magnets is moved inthe circumferential direction of the rotor yoke to adjust coggingtorque, and furthermore the permanent magnets are held in respectiveadjustment positions by the magnet holding member. In the drawing, thedirection in which the magnet is moved is indicated by an arrow.

Referring to FIG. 2, a second embodiment according to the presentinvention will be described. FIG. 2 shows a schematic cross-sectionalview of a permanent magnet motor in a plane passing through the axis ofthe rotor, according to the second embodiment. It should be noted that,in addition to the rotor, a stator 20 having salient poles that aredisposed to be opposed to the magnets also is shown in FIG. 2. In thepermanent magnet motor according to the second embodiment, in contrastto the first embodiment, one (M1) of the permanent magnets is moved inthe radial direction of the rotor yoke to adjust the cogging torque.When moving the permanent magnet in the radial direction, it ispreferable to move the permanent magnet in the radial direction byinserting a non-magnetic shim 16 having a calculated thickness betweenthe permanent magnet (M1) and the rotor yoke 11 and then hold thepermanent magnet with the magnet holding member 13. In the drawing, thedirection in which the magnet is moved is indicated by an arrow.

Referring to FIG. 3, a third embodiment of the present invention isdescribed. FIG. 3 shows a schematic cross-sectional view of a permanentmagnet motor in a plane passing through the axis of the rotor, accordingto the third embodiment. It should be noted that, in addition to therotor, a stator yoke having salient poles that are disposed to beopposed to magnets also is shown in FIG. 3. In the permanent magnetmotor according to the third embodiment, in contrast to the firstembodiment, one (M1) of the permanent magnets is moved in the axialdirection of the rotor yoke 11 to adjust the cogging torque. In thedrawing, the direction in which the magnet is moved is indicated by anarrow.

Regarding the moving direction, a direction in which the smallest numberof magnets is to be moved is chosen based on the results of calculationsusing the mathematical programming method. In the first to thirdembodiments, the permanent magnet is moved only in the circumferentialdirection, the radial direction, or the axial direction to adjust thecogging torque. However, it is also possible to combine these directionsto adjust the cogging torque. Which direction of the circumferentialdirection, the radial direction, the axial direction, and a combinationof these directions is chosen as the moving direction can be determinedin the following manner. That is to say, when the amount of movement isobtained for each direction using the mathematical programming methodand a direction of adjustment in which a small number of magnets are tobe adjusted is used as the moving direction, then an adjustment can bemade with less error. When the value of the cogging torque after anadjustment does not satisfy the target, a readjustment can be made tothe state after the adjustment, using the same method. The direction ofthe readjustment may be a direction different from the direction of theprevious adjustment, and the moving direction is not limited to onlyeither one of these directions.

Regarding the upper limit of the amount of movement of a permanentmagnet, when the permanent magnet is moved in the circumferentialdirection, the permanent magnet can be moved until it touches anadjacent permanent magnet. When the permanent magnet is moved in theradial direction, it is preferable that the permanent magnet can bemoved until an air gap between the inner surface of the stator and thepermanent magnet reaches a minimum of 0.1 mm, since if the stator andthe permanent magnet come into contact with each other, the rotor willno longer rotate. When the permanent magnet is moved in the axialdirection, the permanent magnet can be moved until it touches theholding mechanism.

Moreover, it is possible to fix the above-described plurality ofpermanent magnets in the adjustment positions with an adhesive. In thiscase, it is possible to remove the magnet holding member after fixingthe permanent magnets. FIG. 4 shows a schematic front view of a rotorwhen viewed from a direction perpendicular to the axis of the rotor,according to a fourth embodiment. As shown in FIG. 4, if after apermanent magnet is moved, an adhesive 17 is applied around each of themagnets and the permanent magnets are fixed, then the magnet holdingsystem may be removed. In this case, the magnet holding system can beused to adjust the togging torque of another permanent magnet motor.

It is preferable that the stator yoke is made by the rotationallaminating method in order to eliminate the magnetic anisotropy of thestator yoke. Moreover, it is preferable that the number P of poles ofthe rotor and the number M of salient stator poles satisfy Expression 1or Expression 2 below. More specifically, it is preferable to use any ofthe combinations of the number P of poles and the number M of salientstator poles shown in Table 1 above. The reason for this is that byusing a permanent magnet motor having a relationship between the numberP of poles of the rotor and the number M of salient stator poles thatsatisfies Expression 1 or Expression 2, the cogging torque can bereduced. Moreover, the present invention provides the same effect evenfor a permanent magnet motor that satisfies Expression 3 below, whichexpresses the relationship in an ordinary concentrated winding motor.

P=6n±2, M=6n  Expression 1

wherein n is an integer of 2 or more,

P=3m±1, M=3m  Expression 2

wherein m is an odd number of 3 or more, and

P=2k, M=3k  Expression 3

wherein k is an integer of 1 or more.

EXAMPLES

Working examples of the present invention will be described below withreference to the attached drawings. The working examples described belowdo not limit the present invention.

As shown in Comparative Example 2, by performing the rotationallaminating method, the influence of the stator yoke can be eliminated(see FIGS. 14 and 15), and in Comparative Example 2, the cogging torquewas 0.0122 Nm. However, as described above, it is desirable that thecogging torque is no greater than 0.5% of the rated torque (no greaterthan 0.01 Nm in the present case), and this was not achieved inComparative Example 2. Thus, in the working examples below, anadjustment of the cogging torque was carried out for the permanentmagnet motor shown in Comparative Example 2.

Working Example 1

A permanent magnet motor according to Working Example 1 had the sameconfiguration as the permanent magnet motor according to ComparativeExample 2 described above, except for items particularly describedbelow, such as the position of the permanent magnets and the magnetholding member (see FIG. 11). It should be noted that a shaft runsthrough the center of the rotor yoke made of a low carbon steel, theshaft being made of the same material. In Working Example 1, thepermanent magnets were held by the magnet holding member after apermanent magnet was moved in the circumferential direction, as shown inFIG. 1. Here, as the magnet holding member, magnet holding yokes havingtap holes and hexagon socket set screws serving as the magnet holderswere used. Moreover, the stator yoke was made by the rotationallaminating method.

First, the amount of change in the cogging torque was measured when thepermanent magnet was moved in the circumferential direction. Morespecifically, the amount of a change in the cogging torque that occurredwhen the magnet M1 in FIG. 11 was moved in the circumferential directionwas measured, and components of the change in the cogging torque wereanalyzed using Fourier analysis as in the above-described comparativeexample. FIG. 5 shows the results of the Fourier analysis in which thewaveform of the amount of the change in the cogging torque when thepermanent magnet was moved in the circumferential direction was dividedinto components of respective orders of the waveform. Moreover, thewaveform of the amount of change in the cogging torque was measured eachtime when the magnet M1, M2, M3, M4, or M5 in FIG. 11 was movedindividually in the same direction by 1.25°. FIG. 6 shows theactually-measured waveforms of the amount of change in the coggingtorque when the permanent magnets in different magnet positions weremoved in the radial direction.

As shown in FIG. 5, the components of the cogging torque that changedwere of orders of multiples of 12, most of which were duodenary, and theamount of change was proportional to the amount of movement. Moreover,as shown in FIG. 6, the phase of the waveform was changed by changingthe position (θ) of the magnet. Thus, it was implied that the coggingtorque can be reduced by adjusting the position of a magnet to be moved,and adjusting the amount of movement of the permanent magnet such thatthe waveform of the cogging torque caused by the permanent magnets iscanceled.

In Working Example 1, in order to adjust the cogging torque, theposition of the magnet to be moved and the amount of movement of thepermanent magnet were determined by solving the above-described linearprogramming problem “minimizing (1) under the condition (2)”. Morespecifically, in Working Example 1, the cogging torque was adjusted bymoving the permanent magnet in the circumferential direction. First,data for the cogging torque before adjustment and data for the amount ofchange in the cogging torque when each permanent magnet was moved in thecircumferential direction were input into the linear program, and theamount of movement of each permanent magnet was calculated so as toreduce the cogging torque. More specifically, calculations wereperformed by inputting the following data with respect to each of theletters in Formulae (1) and (2).

In Formula (1), B=10, C₀=0, 0.0001, 0.001, 0.01, and C₁=1. In Formula(2), i=1 to 600, tole=0.001, T_(i)=the waveform of the torque shown inFIG. 15 at the measurement point i (T₁=−0.01268, T₂=−0.01285,T₃=−0.01346, . . . ), and a_(ij)=the waveform of the amount of change inthe cogging torque shown in FIG. 6 divided by an amount of movement of1.25° (a₁₁=0.00203, a₂₁=0.00195, a₃₁=0.00184, a₁₂=0.00043, a₂₂=0.00023,. . . ).

Table 2 shows the amount of movement (X_(i)) of the magnets and thecogging torque at each value of C₀/C₁.

TABLE 2 Coefficients C₀/C₁ 0.01 0.001 0.0001 0 Amount of X₁ 0 0 0 0.3Adjustment (°) X₂ 0 0 0.3 0.7 X₃ 0 1 0.5 0.3 X₄ 0 2.2 2.8 3.2 X₅ 0 0 0 0X₆ 0 0 0 0 X₇ 0 0 0 0 X₈ 0 0 0 0 X₉ 0 0 0 0 X₁₀ 0 0 0 0 Cogging torquebefore 0.0122 0.0122 0.0122 0.0122 adjustment (Nm) Cogging torque after0.0122 0.0036 0.0036 0.0036 adjustment (Nm)

From the calculation results, it was found that the cogging torque canbe optimized by moving the magnet M3 by 1.0° and the magnet M4 by 2.2°in the same circumferential direction. According to the calculationresults, the permanent magnets were moved, and as in the comparativeexample, the waveform of the cogging torque was measured and thecomponents of the cogging torque that changed were analyzed usingFourier analysis. FIG. 7 shows the actually-measured waveform of thecogging torque of the permanent magnet motor according to WorkingExample 1. FIG. 8 shows the results of the Fourier analysis in which thewaveform of the cogging torque according to Comparative Example 2 andWorking Example 1 was divided into components of respective orders ofthe waveform. As shown in FIG. 8, the duodenary component (peak value)was 0.0053 Nm in Comparative Example 2, whereas it was significantlyreduced to 0.0004 Nm in Working Example 1. Moreover, as shown in FIG. 7,the cogging torque was 0.0122 Nm overall in Comparative Example 2,whereas it became 0.0036 Nm in Working Example 1. In this way, thetarget cogging torque of no greater than 0.01 Nm (no greater than 0.5%of the rated torque) could be achieved in Working Example 1.

Working Example 2

Working Example 2 was the same as Working Example 1 except that thepermanent magnet was moved in the radial direction as shown in FIG. 2.First, the amount of change in the cogging torque when the permanentmagnet was moved in the radial direction was measured. Non-magnetic SUSwas used for the non-magnetic shim. More specifically, the amount ofchange in the cogging torque that occurred when the magnet M1 in FIG. 11was moved in the radial direction was measured, and as in theabove-described comparative example, the components of the coggingtorque that changed were analyzed using Fourier analysis. FIG. 9 showsthe results of the Fourier analysis in which the waveform of the amountof change in the cogging torque when the permanent magnet was moved inthe radial direction was divided into components of respective orders ofthe waveform. As in the case where the permanent magnet was moved in thecircumferential direction, the components of the cogging torque thatchanged were of orders of multiples of 12, most of which were duodenary.Moreover, the amount of change in the cogging torque was proportional tothe amount of movement. From these results, it was implied that evenwhen the permanent magnet is moved in the radial direction, the amountthat the permanent magnet is to be moved in the radial direction inorder to reduce the cogging torque can be obtained using the linearprogramming method, as in Working Example 1.

As in Working Example 1, data for the cogging torque before adjustmentand data for the amount of change in the cogging torque when eachpermanent magnet was moved in the radial direction were input into thelinear program, and the amount of movement of each permanent magnet wascalculated so as to reduce the cogging torque. From the calculationresults, it was found that the cogging torque could be optimized bymoving the magnets M1 and M6 by 0.52 mm, the magnets M4 and M9 by 0.26mm, and the magnets M5 and M10 by 0.22 mm in the radial direction.According to the calculation results, the permanent magnets were moved,and as in the comparative example, the waveform of the cogging torquewas measured and the components of the cogging torque that changed wereanalyzed using Fourier analysis. The duodenary component (peak value)was 0.0053 Nm in Comparative Example 2, whereas it was significantlyreduced to 0.0018 Nm in Working Example 2. Moreover, the overall coggingtorque was 0.0122 Nm in Comparative Example 2, whereas it became 0.0068Nm in Working Example 2. In this way, the target cogging torque of nogreater than 0.01 Nm (no greater than 0.5% of the rated torque) could beachieved in Working Example 2.

Working Example 3

Working Example 3 was the same as Working Example 1 except that thepermanent magnet was moved in the axial direction as shown in FIG. 3.First, the amount of change in the cogging torque when the permanentmagnet was moved in the axial direction was measured. More specifically,the amount of change in the cogging torque that occurred when the magnetM1 shown in FIG. 11 was moved in the axial direction was measured, andthe components of the cogging torque that changed were analyzed usingFourier analysis. FIG. 10 shows the results of the Fourier analysis inwhich the waveform of the amount of change in the cogging torque whenthe permanent magnet was moved in the axial direction was divided intocomponents of respective orders of the waveform. As in the cases wherethe permanent magnet was moved in the circumferential direction or inthe radial direction, almost all of the components of the cogging torquewere of duodenary order. Moreover, the amount of change in the coggingtorque was proportional to the amount of movement. Accordingly, it wasimplied that even when the permanent magnet is moved in the axialdirection, the amount that the permanent magnet is to be moved in theaxial direction in order to reduce the cogging torque could be obtainedusing the linear programming method, as in Working Examples 1 and 2.

As in Working Example 1, data for the cogging torque before adjustmentand data for the amount of change in the cogging torque when eachpermanent magnet was moved in the axial direction were input into thelinear program, and the amount of movement of each permanent magnet wascalculated so as to reduce the cogging torque. From the calculationresults, it was found that the cogging torque can be optimized by movingthe magnet M2 by 2.3 mm and the magnets M3 and M8 by 3.8 mm in the axialdirection. According to the calculation results, the permanent magnetswere moved, and as in the comparative example, the waveform of thecogging torque was measured and the components of the cogging torquethat changed were analyzed using Fourier analysis. The duodenary ordercomponent (peak value) was 0.0053 Nm in Comparative Example 2, whereasit was significantly reduced to 0.0017 Nm in Working Example 3.Moreover, the overall cogging torque was 0.0122 Nm in ComparativeExample 2, whereas it became 0.0064 Nm in Working Example 3. In thisway, the target cogging torque of no greater than 0.01 Nm (no greaterthan 0.5% of the rated torque) could be achieved in Working Example 3.

Having thus described certain embodiments of the present invention, itis to be understood that the invention defined by the appended claims isnot to be limited by particular details set forth in the abovedescription as many apparent variations thereof are possible withoutdeparting from the spirit or scope thereof as hereinafter claimed. Thefollowing claims are provided to ensure that the present applicationmeets all statutory requirements as a priority application in alljurisdictions and shall not be construed as setting forth the full scopeof the present invention.

1. A method for adjusting a cogging torque of a permanent magnet motor,the permanent magnet motor comprising: a rotor comprising a rotor yokeand a plurality of permanent magnet that are disposed on a side face ofthe rotor yoke at predetermined intervals such that polarities of thepermanent magnets alternate in a circumferential direction of the rotoryoke; and a stator comprising a stator yoke that is disposed at adistance from the rotor, salient magnetic poles that are disposed on thestator yoke at even intervals with respect to the circumferentialdirection and that are opposed to the permanent magnets, and three-phaseconnected armature windings that are concentratedly wound around therespective salient magnetic poles, wherein the method comprises stepsof: disposing the plurality of permanent magnets in reference positionsthat are located at even intervals with respect to the circumferentialdirection of the rotor yoke, are equidistant from the central axis withrespect to a radial direction, and are equidistant from axial directionends of the rotor yoke with respect to an axial direction, such that thepolarities of the permanent magnets alternate in the circumferentialdirection; and moving at least one of the permanent magnets in at leastone of the circumferential, radial, and axial directions of the rotoryoke to adjust the cogging torque.
 2. The method according to claim 1,wherein the step of adjusting the cogging torque comprises: measuringthe cogging torque before adjusting the cogging torque; measuring thecogging torque when one of the permanent magnets is moved in at leastone of the circumferential, radial, and axial directions of the rotoryoke; and determining a permanent magnet to be moved and an amount thatthe permanent magnet is to be moved based on obtained measurementvalues, using a linear programming method, such that the cogging torquecan be more reduced.
 3. The method according to claim 2, furthercomprising: holding the plurality of permanent magnets in the referencepositions by magnet holding members; and/or holding the plurality ofpermanent magnets in adjustment positions to which the permanent magnetshave been moved, by magnet holding members.
 4. The method according toclaim 3, further comprising: fixing the plurality of permanent magnetsin the respective adjustment positions by an adhesive; and removing themagnet holding member after fixing the permanent magnets.
 5. The methodaccording to claim 1, wherein the linear programming method minimizes(1) under a condition (2): $\begin{matrix}{Z = {{C_{0}{\sum\limits_{j = 1}^{B}\; X_{j}}} + {C_{1}W}}} & (1) \\\left. \begin{matrix}{{{\sum\limits_{j = 1}^{B}\; {a_{ij}X_{j}}} + \frac{W}{2}} \geq {{- T_{i}} - {\frac{1}{2}{tole}} + T_{l}}} \\{{{- {\sum\limits_{j = 1}^{B}\; {a_{ij}X_{j}}}} + \frac{W}{2}} \geq {T_{i} - {\frac{1}{2}{tole}} - T_{l}}} \\{X_{j},{W \geq 0}}\end{matrix} \right\} & (2)\end{matrix}$ wherein B represents a number of magnets, i represents apoint at which the cogging torque is measured, j represents a positionof the magnet, X_(j) is an amount of movement of a j-th magnet, T_(i) isthe cogging torque at an i-th point before adjusting the cogging torque,a_(ij) is an amount of change in the cogging torque at the i-th pointper amount of movement of the j-th magnet when the j-th magnet is moved,T₁ is a loss torque before adjusting the cogging torque, tole is atarget cogging torque, W is a difference between the target coggingtorque and a calculated value of the cogging torque after the permanentmagnet has been moved, and C₀ and C₁ are coefficients that are setaccording to a balance between the amount of movement of the magnet andthe cogging torque.
 6. The method according to claim 4, wherein thelinear programming method minimizes (1) under a condition (2):$\begin{matrix}{Z = {{C_{0}{\sum\limits_{j = 1}^{B}\; X_{j}}} + {C_{1}W}}} & (1) \\\left. \begin{matrix}{{{\sum\limits_{j = 1}^{B}\; {a_{ij}X_{j}}} + \frac{W}{2}} \geq {{- T_{i}} - {\frac{1}{2}{tole}} + T_{l}}} \\{{{- {\sum\limits_{j = 1}^{B}\; {a_{ij}X_{j}}}} + \frac{W}{2}} \geq {T_{i} - {\frac{1}{2}{tole}} - T_{l}}} \\{X_{j},{W \geq 0}}\end{matrix} \right\} & (2)\end{matrix}$ wherein B represents a number of magnets, i represents apoint at which the cogging torque is measured, j represents a positionof the magnet, X_(j) is an amount of movement of a j-th magnet, T_(i) isthe cogging torque at an i-th point before adjusting the cogging torque,a_(ij) is an amount of change in the cogging torque at the i-th pointper amount of movement of the j-th magnet when the j-th magnet is moved,T₁ is a loss torque before adjusting the cogging torque, tole is atarget cogging torque, W is a difference between the target coggingtorque and a calculated value of the cogging torque after the permanentmagnet has been moved, and C₀ and C₁ are coefficients that are setaccording to a balance between the amount of movement of the magnet andthe cogging torque.